# Why aren't repeating decimals irrational but something like $\pi$ is?

We use closest representations for both of them, but they are not completely true.

$\frac{22}7$ and $3.14$ are not exactly $\pi$ but we use them as the best option available.

$\frac13$ is $0.\bar3$ but that can be $0.333$ or $0.333333$ and these are not equal.

So why is one irrational and other is not?

• What do you mean when you say that $\frac13$ can be $0.333$ or $0.333333$? Neither of those is $\frac13$. Jun 6, 2018 at 3:12
• The best approximation depends on context. There is nothing inherently “best” about 3.14 (or any other approximation) as an approximation for $\pi$... Jun 6, 2018 at 3:15
• You could ask "why are repeating decimals rational?". Answer: the formula for the sum of an infinite geometric series. Jun 6, 2018 at 3:16
• Harveen, $1/3$ has an infinite decimal approximation but that does not mean it is irrational. The defining property of being irrational is that it can't be written as a fraction. Since $1/3$ can be written as a fraction, it is by definition rational. One thing you might want to consider: $1/3$ is actually equal to $.1$ in base $3$, but $\pi$ will have an infinite expansion no matter what base you pick. Jun 6, 2018 at 3:19
• @HarveenBhatia: No. Have you learned about writing numbers in different bases? The fraction $1/3$ can be written as $.1$ in base $3$. It does not go on forever in that base. Jun 6, 2018 at 4:04

Repeating decimals are simply geometric series which add up to

$$a + ar + ar^2 + \cdots = \frac {a}{1-r}$$

where $a$ and $r$ are rational numbers, so the result is rational.

For example:

\begin{align} 0.23\,23\,23\ldots &= 0.23 +0.00\,23+0.00\,00\,23 +\cdots \\[2ex] &= 0.23 + 0.23\left(\frac1{100}\right) + 0.23\left(\frac1{100}\right)^2 + \cdots \\[2ex] &= \frac{0.23}{1-\frac1{100}}=\frac {23}{99} \end{align}

• Brilliant way to think about it Jun 6, 2018 at 3:52
• I love this answer a lot [+1]!! I really hope you don’t mind that I took the opportunity to provide a more robust explanation of the arithmetic you performed. Jun 6, 2018 at 19:30
• @ChaseRyanTaylor Thanks for the informational editing. I liked it a lot. Jun 6, 2018 at 20:05
• I have undone the down-vote by giving you an up-vote. Some members seem to forget there is no lower age limit or lower experience limit for membership here. And the theory of $\Bbb R$ is often taught very poorly to the lower age-levels in schools. Jun 7, 2018 at 1:37
• @DanielWainfleet Thanks for your attention. I appreciate your up-vote. Jun 7, 2018 at 4:12

The word irrational means the number cannot be expressed in a ratio of two integers. $\frac{1}{3}$ can obviously be expressed as a ratio of two integers. For $\pi$ and other irrational numbers however, there are no integers $a$ and $b$ where $\frac{a}{b}$ will ever equal that number.

• Good point. I suggest you reference the full definition of irrational numbers to make your answer more complete. Something like this: mathworld.wolfram.com/IrrationalNumber.html Jun 6, 2018 at 3:37
• Thanks for the link! Jun 6, 2018 at 3:41

Having an infinite decimal expansion is not what makes a number irrational. A rational number is any number that can be expressed as a fraction - that is, the words rational number and fraction are essentially synonymous.

$\pi$ is irrational because it cannot be expressed as a fraction - $22/7$ is a close approximation, but no fraction can ever exactly represent $\pi$. On the other hand, $1/3$, being a fraction, is rational by definition, irrespective of any infinite repeating decimal sequence.

The connection between irrational numbers and decimal sequences is this - if a number is irrational, it's decimal sequence cannot terminate, and furthermore the decimal sequence cannot be periodic, or repeating. This doesn't mean there can't be any pattern in the digits, just that they can't repeat themselves endlessly in uninterrupted fashion. A rational number, on the other hand, can have either a finite decimal expansion or an infinite expansion, but if the decimal expansion for a rational number is infinite, then it must be periodic, or repeating.

So the properties of the decimal expansions of rational and irrational number are a consequence of their definition in terms of representability by fractions, and not a definition in and of themselves.

Something is irrational if its decimals go on forever and do so with no pattern. The number $\pi$ fits both of these criteria; however $1/3=0.333\cdots$ does not fit the second criterion because its digits repeat.

If numbers exhibit a pattern, it’s a clue to us that our decimal (base $10$) system represents them cleanly—and sure enough, when we look at the math, we see this is the case!

• $0.1010010001000010000010000001\ldots$ exhibits a pretty clear pattern, but it's not rational.
– user14972
Jun 6, 2018 at 3:32
• @Hurkyl “it’s a clue” — not the end-all-be-all. If you want to open another Pandora’s box for the OP, who is still cultivating her number sense, then I challenge you to also offer a resolution to that. Jun 6, 2018 at 3:45
• The trick with clues is to be careful about what precisely you're cluing. You know and I know that, when you use the word "pattern", you specifically mean "repeating decimal". The problem is that she who is "still cultivating her number sense" is among the people most likely to think you are using "pattern" to mean "pattern". You're offering the OP extra chances to get the wrong idea lodged in her head, and I don't think the imprecise choice of words is actually any simpler or clearer than a more precise choice.
– user14972
Jun 6, 2018 at 3:54

"Rationality" is not about closeness of approximations - we use approximations for everything! If a person weighs $151.6629943$ pounds, they'll probably say they weigh $151$. And $151.6629943$ doesn't even go on forever!

A number is "rational" if it is a whole number divided by another whole number. Repeating decimals can always be written that way - if the repeating section is $a_1a_2\cdots a_k$, then the number can be written as $\frac{a_1a_1\cdots a_k}{99\cdots 9}$. Importantly, this fraction representation is required to be exact, unlike the decimal approximations $0.333$ and $0.3333333$ you were talking about.

$\pi$, on the other hand, cannot be written this way. $\frac{22}{7}$ is a good approximation, but it is not exact; in fact, there is no fraction that would exactly equal $\pi$.